A353092 Inventory sequence counting prime factors. (See comment.)

0, 1, 0, 3, 1, 0, 5, 2, 0, 6, 3, 1, 0, 8, 4, 2, 1, 0, 10, 5, 3, 1, 0, 12, 7, 3, 2, 0, 13, 11, 3, 2, 0, 14, 14, 5, 2, 0, 15, 16, 6, 2, 1, 0, 17, 18, 7, 3, 1, 0, 19, 21, 8, 4, 1, 0, 21, 21, 11, 4, 1, 0, 23, 23, 12, 5, 1, 0, 25, 25, 14, 5, 1, 0, 27, 26, 16, 6, 2

Offset: 0

Author

David James Sycamore, Apr 22 2022

Keywords

• nonn

Comments

0 and 1 are the only nonnegative integers which have no prime factors. The sequence uses this property as follows: Record the number of existing terms which have 0 prime factors, then the number having 1 prime factor, then 2, 3 and so on until reaching a number k such that there are no terms having k prime factors (counted with multiplicity). At this point record a 0 term, and restart the count.

Examples

			a(0) = 0 since at first there are no terms, hence 0 terms with 0 prime factors. The count now restarts because a 0 term has occurred.
a(1) = 1 because now there is one term (a(0)) which has no prime factor.
a(2) = 0 because there is no term with one factor. The count now restarts.
a(3) = 3 because all three prior terms have no prime factor.
a(4) = 1 since a(3) is prime, the first to occur in the sequence.
a(5) = 0 since there are no terms with 2 prime divisors. The count now restarts...
As an irregular table the sequence starts:
   0;
   1, 0;
   3, 1, 0;
   5, 2, 0;
   6, 3, 1, 0;
   8, 4, 2, 1, 0;
  10, 5, 3, 1, 0;
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..11196 (as an irregular table, rows m = 0..2^10, flattened)
• Michael De Vlieger, Scatterplot of a(n), n = 1..11185 (2^10 zeros), with a color code assigning black to 0, and color-coding trajectories, starting with red and ending with magenta, pertaining to omega(k), 1 <= k <= 15 respectively.
• Michael De Vlieger, Log-log scatterplot of a(n), n = 1..11185 (2^10 zeros), color-coding trajectories, starting with red and ending with magenta, pertaining to omega(k), 1 <= k <= 15 respectively.
• Index entries for sequences related to the inventory sequence

Crossrefs

Cf. A342585, A347791.

Programs

Mathematica

Block[{a, c, j, k, m}, a[1] = c[] = 0; j = c[-1] = c[0] = 1; Do[k = 0; While[c[k] > 0, j++; Set[m, c[k]]; Set[a[j], m]; c[If[m < 2, 0, PrimeOmega[m]]]++; k++]; j++; Set[a[j], 0]; c[0]++, 16]; Array[a, j] ] (* _Michael De Vlieger, Apr 23 2022 *)

Python

from sympy import factorint
from collections import Counter
def f(n): return 0 if n < 2 else sum(e for p, e in factorint(n).items())
def aupton(nn):
    num, alst, inventory = 0, [0], Counter([0])
    for n in range(1, nn+1):
        c = inventory[num]
        num = 0 if c == 0 else num + 1
        alst.append(c)
        inventory.update([f(c)])
    return alst
print(aupton(78)) # Michael S. Branicky, Apr 22 2022

Extensions

a(50) and beyond from Michael S. Branicky, Apr 22 2022